The sum of two numbers is $112$, and their difference is $46$. What are the two numbers?
Solution: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 112}$ ${x-y = 46}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 158 $ $ x = \dfrac{158}{2} $ ${x = 79}$ Now that you know ${x = 79}$ , plug it back into $ {x+y = 112}$ to find $y$ ${(79)}{ + y = 112}$ ${y = 33}$ You can also plug ${x = 79}$ into $ {x-y = 46}$ and get the same answer for $y$ ${(79)}{ - y = 46}$ ${y = 33}$ Therefore, the larger number is $79$, and the smaller number is $33$.